第二次作业

机器学习

叶雨菲 计32 2013011325

5.4
$error_D \in [0.2, 0.6]$
$\therefore p = 0.4$
要求$P(\frac{|x-\mu|}{\sigma} < a ) > 95\%$
$\therefore a = z_{0.975} = 1.96$
$\therefore |x-\mu| < 1.96 \sqrt{p(1-p)/n}$
$\therefore 0.05 < 1.96 \sqrt{0.21/n}$
$\therefore n > 322.69$
$\therefore n = 323$

Compare results on 5% & 100% training set respectively

1. Estimations of errorD and the C.I. respectively
2. What’s the confidence of algorithm A is better than B in general?

得到30组数据，计算期望和标准差，得：
$error_{S5\%} = 100-82.725=17.275 , error_{S100\%} = 100-83.085=16.915$
$\sigma_{5\%} = 0.00814 , \sigma_{100\%} = 0.00105$
分别估计$error_D$和置信区间
我全部使置信区间为two-side 95%
对于5% $error_D = error_S \pm z_{0.95}\sigma = 82.72 \pm 0.016$
对于100% $error_D = error_S \pm z_{0.95}\sigma = 83.09 \pm 0.0021$

2
when algorithm A is better than B
$P(error_{D_B} - error_{D_A} > 0) = P(error_{S_B} - error_{S_A} > 0)$
其中$\sigma = \sqrt{\sigma_B^2 + \sigma_A^2} = 0.0082$
$d = error_{S_B} - error{S_A}$服从$\mu = -0.36, \sigma = 0.0082$的高斯分布
$P((d + 0.36)/0.0082 > (0+0.36)/0.0082 ) = P(X > -43)\approx 1$

所以说，在我的Naive Bayse classification中，取数据集取100% 的算法几乎100%肯定的比取训练集为5%的要好